Time allowed: **2 h 0 min**.

The total mark for this paper is **100**.

The marks for each question are shown in brackets **[ ]**.

\[g(x) = \dfrac{2x+5}{x-3} \qquad x ⩾ 5 \]

(a) Find \(gg(5)\).

(b) State the range of \(g\).

(c) Find \(g^{-1}(x) \), stating its domain.

(a) Find \(gg(5)\).

[2]

(b) State the range of \(g\).

[1]

(c) Find \(g^{-1}(x) \), stating its domain.

[3]

Composite functions Inverse functions Functions, mappings, domain and range

Relative to a fixed origin *O*,

the point*A* has position vector \((2\bold{i}+3\bold{j}-4\bold{k})\),

the point*B* has position vector \((4\bold{i}-2\bold{j}+3\bold{k})\),

and the point*C* has position vector \((a\bold{i}+5\bold{j}-2\bold{k})\), where *a* is a constant and \({a < 0}\)

*D* is the point such that \(\overrightarrow{AB} = \overrightarrow{BD} \).

(a) Find the position vector of*D*.

Given \(|\overrightarrow{AC}| = 4\)

(b) find the value of*a*.

the point

the point

and the point

(a) Find the position vector of

[2]

Given \(|\overrightarrow{AC}| = 4\)

(b) find the value of

[3]

Magnitude and direction of a vector Position vectors

(a) “If *m* and *n* are irrational numbers, where \(m≠n\), then *mn* is also irrational.”

**Disprove** this statement by means of a counter example.

(b) (i) Sketch the graph of \(y=|x|+3\)

(ii) Explain why \(|x|+3 ⩾ |x+3|\) for all real values of*x*.

[2]

(b) (i) Sketch the graph of \(y=|x|+3\)

(ii) Explain why \(|x|+3 ⩾ |x+3|\) for all real values of

[3]

Disproof by counter example Modulus

(i) Show that

\[\displaystyle\sum^{16}_{r=1} (3 + 5r + 2^r) = 131 798 \]

(ii) A sequence \(u_1, u_2, u_3, ... \) is defined by

\[ u_{n+1} = \dfrac{1}{u_n}, \qquad u_1 = \dfrac{2}{3} \]

Find the exact value of \(\displaystyle\sum^{100}_{r=1} u_r \)

\[\displaystyle\sum^{16}_{r=1} (3 + 5r + 2^r) = 131 798 \]

[4]

(ii) A sequence \(u_1, u_2, u_3, ... \) is defined by

\[ u_{n+1} = \dfrac{1}{u_n}, \qquad u_1 = \dfrac{2}{3} \]

Find the exact value of \(\displaystyle\sum^{100}_{r=1} u_r \)

[3]

Sequences Sigma notation Arithmetic sequences and series Geometric sequences and series

The equation \(2x^3+x^2-1=0 \) has exactly one real root.

(a) Show that, for this equation, the Newton-Raphson formula can be written

\[x_{n+1} = \dfrac{4x_n^3 + x_n^2 + 1}{6x_n^2 + 2x_n} \]

Using the formula given in part (a) with \(x_1=1\)

(b) find the values of \(x_2\) and \(x_3\)

(c) Explain why, for this question, the Newton-Raphson method cannot be used with \({x_1=0}\)

(a) Show that, for this equation, the Newton-Raphson formula can be written

\[x_{n+1} = \dfrac{4x_n^3 + x_n^2 + 1}{6x_n^2 + 2x_n} \]

[3]

Using the formula given in part (a) with \(x_1=1\)

(b) find the values of \(x_2\) and \(x_3\)

[2]

(c) Explain why, for this question, the Newton-Raphson method cannot be used with \({x_1=0}\)

[1]

\[f(x)=-3x^3+8x^2-9x+10, \qquad x∈ℝ\]

(a) (i) Calculate \(f(2)\)

(ii) Write \(f(x)\) as a product of two algebraic factors.

Using the answer to (a)(ii),

(b) prove that there are exactly two real solutions to the equation

\[-3y^6+8y^4-9y^2+10=0 \]

(c) deduce the number of real solutions, for \({7π⩽θ<10π}\), to the equation

\[3\tan^3{θ}-8\tan^2{θ}+9\tan{θ}-10=0\]

(a) (i) Calculate \(f(2)\)

(ii) Write \(f(x)\) as a product of two algebraic factors.

[3]

Using the answer to (a)(ii),

(b) prove that there are exactly two real solutions to the equation

\[-3y^6+8y^4-9y^2+10=0 \]

[2]

(c) deduce the number of real solutions, for \({7π⩽θ<10π}\), to the equation

\[3\tan^3{θ}-8\tan^2{θ}+9\tan{θ}-10=0\]

[1]

Discriminant of a quadratic function Polynomials Factor theorem Solve trigonometric equations

(i) Solve, for \({0⩽x⩽\dfrac{π}{2}}\), the equation

\[4\sin{x} = \sec{x} \]

(ii) Solve, for \({0⩽x⩽360°}\), the equation

\[5\sin{θ} - 5\cos{θ} = 2 \]

giving your answers to one decimal place.

(*Solutions based entirely on graphical or numerical methods are not acceptable.*)

\[4\sin{x} = \sec{x} \]

[4]

(ii) Solve, for \({0⩽x⩽360°}\), the equation

\[5\sin{θ} - 5\cos{θ} = 2 \]

giving your answers to one decimal place.

(

[5]

a cosθ + b sinθ Solve trigonometric equations

Figure 1 is a graph showing the trajectory of a rugby ball.

The height of the ball above the ground,

The ball travels in a vertical plane.

The ball reaches a maximum height of 12 metres and hits the ground at a point 40 metres from where it was kicked.

(a) Find a quadratic equation linking

[3]

The ball passes over the horizontal bar of a set of rugby posts that is perpendicular to the path of the ball. The bar is 3 metres above the ground.

(b) Use your equation to find the greatest horizontal distance of the bar from

[3]

(c) Give one limitation of the model.

[1]

Quadratic functions and graphs Use of functions in modelling

Given that *θ* is measured in radians, prove, from first principles, that

\[\dfrac{d}{dθ}(\cos{}) = -\sin{θ} \]

You may assume the formula for \(\cos{(A±B)}\) and that as \({h→0}\), \(\dfrac{\sin{h}}{h} → 1 \) and \(\dfrac{\cos{h} - 1}{h} → 0 \)

\[\dfrac{d}{dθ}(\cos{}) = -\sin{θ} \]

You may assume the formula for \(\cos{(A±B)}\) and that as \({h→0}\), \(\dfrac{\sin{h}}{h} → 1 \) and \(\dfrac{\cos{h} - 1}{h} → 0 \)

Differentiation from first principles

A spherical mint of radius 5 mm is placed in the mouth and sucked.

Four minutes later, the radius of the mint is 3 mm.

In a simple model, the rate of decrease of the radius of the mint is inversely proportional to the square of the radius.

Using this model and all the information given,

(a) find an equation linking the radius of the mint and the time.

(You should define the variables that you use.)

(b) Hence find the total time taken for the mint to completely dissolve. Give your answer in minutes and seconds to the nearest second.

(c) Suggest a limitation of the model.

Four minutes later, the radius of the mint is 3 mm.

In a simple model, the rate of decrease of the radius of the mint is inversely proportional to the square of the radius.

Using this model and all the information given,

(a) find an equation linking the radius of the mint and the time.

(You should define the variables that you use.)

[5]

(b) Hence find the total time taken for the mint to completely dissolve. Give your answer in minutes and seconds to the nearest second.

[2]

(c) Suggest a limitation of the model.

[1]

Differential equations Solve differential equations

\[\dfrac{1+11x-6x^2}{(x-3)(1-2x)} ≡ A + \dfrac{B}{(x-3)} + \dfrac{C}{(1-2x)} \]

(a) Find the values of the constants*A*, *B* and *C*.

\[f(x) = \dfrac{1+11x-6x^2}{(x-3)(1-2x)}, \qquad x > 3\]

(b) Prove that \(f(x)\) is a decreasing function.

(a) Find the values of the constants

[4]

\[f(x) = \dfrac{1+11x-6x^2}{(x-3)(1-2x)}, \qquad x > 3\]

(b) Prove that \(f(x)\) is a decreasing function.

[3]

Partial fractions Differentiation of standard functions Turning points

(a) Prove that

\[1-\cos{2θ} ≡ \tan{θ}\sin{2θ}, \]\[θ ≠ \dfrac{(2n+1)π}{2}, \qquad n∈Z\]

(b) Hence solve, for \({-\dfrac{π}{2} < x < \dfrac{π}{2}} \), the equation

\[(\sec^2{x}-5)(1-\cos{2x}) = 3\tan^2{x}\sin{2x} \]

Give any non-exact answer to 3 decimal places where appropriate.

\[1-\cos{2θ} ≡ \tan{θ}\sin{2θ}, \]\[θ ≠ \dfrac{(2n+1)π}{2}, \qquad n∈Z\]

[3]

(b) Hence solve, for \({-\dfrac{π}{2} < x < \dfrac{π}{2}} \), the equation

\[(\sec^2{x}-5)(1-\cos{2x}) = 3\tan^2{x}\sin{2x} \]

Give any non-exact answer to 3 decimal places where appropriate.

[6]

Solve trigonometric equations Proofs involving trigonometry

Figure 2 shows a sketch of part of the curve

\[y = x\ln{x}, \qquad x > 0\]

The line

The region

Show that the exact area of

Straight lines Area under and between curves Integration by parts

A scientist is studying a population of mice on an island.

The number of mice,*N*, in the population, *t* months after the start of the study, is modelled by the equation

\[N = \dfrac{900}{3+7e^{-0.25t}},\]\[t∈ℝ, \qquad t⩾0\]

(a) Find the number of mice in the population at the start of the study.

(b) Show that the rate of growth \(\dfrac{dN}{dt}\) is given by \({\dfrac{dN}{dt} = \dfrac{N(300-N)}{1200}}\)

The rate of growth is a maximum after*T* months.

(c) Find, according to the model, the value of*T*.

According to the model, the maximum number of mice on the island is*P*.

(d) State the value of*P*.

The number of mice,

\[N = \dfrac{900}{3+7e^{-0.25t}},\]\[t∈ℝ, \qquad t⩾0\]

(a) Find the number of mice in the population at the start of the study.

[1]

(b) Show that the rate of growth \(\dfrac{dN}{dt}\) is given by \({\dfrac{dN}{dt} = \dfrac{N(300-N)}{1200}}\)

[4]

The rate of growth is a maximum after

(c) Find, according to the model, the value of

[4]

According to the model, the maximum number of mice on the island is

(d) State the value of

[1]

Exponential modelling Quotient rule